Remark 2.7 The structure of the aforementioned concentration inequality is inspired by the work in [1]} on bounding deviations of empirical measure formed by interacting particle systems from their infinite particle limit (Mckean-Vlasov diffusion); see e.g., problem section in [2]} for an involved discussion on this matter. However, there is a subtle difference in our approach – if we work with an \(\ell ^{2}\)  inspired metric on \(\mathcal {X}^{N}\)  as in [1]}, i.e.,
\(d_{(N)} ^2 (x^N,y^N):= \sqrt{\sum _{i=1}^{N} d^2(x_i,y_i)}.\) 
